MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cntzidss Unicode version

Theorem cntzidss 15063
Description: If the elements of  S commute, the elements of a subset  T also commute. (Contributed by Mario Carneiro, 25-Apr-2016.)
Hypothesis
Ref Expression
cntzmhm.z  |-  Z  =  (Cntz `  G )
Assertion
Ref Expression
cntzidss  |-  ( ( S  C_  ( Z `  S )  /\  T  C_  S )  ->  T  C_  ( Z `  T
) )

Proof of Theorem cntzidss
StepHypRef Expression
1 simpr 448 . 2  |-  ( ( S  C_  ( Z `  S )  /\  T  C_  S )  ->  T  C_  S )
2 simpl 444 . . 3  |-  ( ( S  C_  ( Z `  S )  /\  T  C_  S )  ->  S  C_  ( Z `  S
) )
3 eqid 2387 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
4 cntzmhm.z . . . . . 6  |-  Z  =  (Cntz `  G )
53, 4cntzssv 15054 . . . . 5  |-  ( Z `
 S )  C_  ( Base `  G )
62, 5syl6ss 3303 . . . 4  |-  ( ( S  C_  ( Z `  S )  /\  T  C_  S )  ->  S  C_  ( Base `  G
) )
73, 4cntz2ss 15058 . . . 4  |-  ( ( S  C_  ( Base `  G )  /\  T  C_  S )  ->  ( Z `  S )  C_  ( Z `  T
) )
86, 7sylancom 649 . . 3  |-  ( ( S  C_  ( Z `  S )  /\  T  C_  S )  ->  ( Z `  S )  C_  ( Z `  T
) )
92, 8sstrd 3301 . 2  |-  ( ( S  C_  ( Z `  S )  /\  T  C_  S )  ->  S  C_  ( Z `  T
) )
101, 9sstrd 3301 1  |-  ( ( S  C_  ( Z `  S )  /\  T  C_  S )  ->  T  C_  ( Z `  T
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    C_ wss 3263   ` cfv 5394   Basecbs 13396  Cntzccntz 15041
This theorem is referenced by:  gsumzres  15444  gsumzf1o  15446  gsumzaddlem  15453  gsumzadd  15454  gsumzsplit  15456  gsumconst  15459  gsumpt  15472  dprdfadd  15505
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-cntz 15043
  Copyright terms: Public domain W3C validator